Disney does not have service that would send you distilled water. There are some other alternatives, however. You could utilize a local service that delivers groceries to WDW hotel rooms. Former Moms Panelist Bret lists those services
HERE. A grocery delivery service would probably work best if you have other items you want delivered as well (snacks, milk, etc) as there are often minimum order requirements.
If it's only distilled water you want. your best bet might be to order it online and have it delivered to yourself at your resort. I just checked on Amazon.com and the have many brands and sizes of distilled water that they will ship anywhere in the US. You don't say which All Star you're staying at, so I can't give you the mailing address. To get this, call 407-W-DISNEY and ask them for the mailing address of your particular hotel. Address the package to yourself at that address. If at all possible, try to include the word GUEST and your arrival date on the package as well.
Of course, if you have access to a car, the simplest thing to do would be to drive to a local area drugstore. The All Star bell services desk can give you directions.